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Learning on the correct class for domain inverse problems of gravimetry

Yihang Chen, Wenbin Li

Abstract

We consider end-to-end learning approaches for inverse problems of gravimetry. Due to ill-posedness of the inverse gravimetry, the reliability of learning approaches is questionable. To deal with this problem, we propose the strategy of learning on the correct class. The well-posedness theorems are employed when designing the neural-network architecture and constructing the training set. Given the density-contrast function as a priori information, the domain of mass can be uniquely determined under certain constrains, and the domain inverse problem is a correct class of the inverse gravimetry. Under this correct class, we design the neural network for learning by mimicking the level-set formulation for the inverse gravimetry. Numerical examples illustrate that the method is able to recover mass models with non-constant density contrast.

Learning on the correct class for domain inverse problems of gravimetry

Abstract

We consider end-to-end learning approaches for inverse problems of gravimetry. Due to ill-posedness of the inverse gravimetry, the reliability of learning approaches is questionable. To deal with this problem, we propose the strategy of learning on the correct class. The well-posedness theorems are employed when designing the neural-network architecture and constructing the training set. Given the density-contrast function as a priori information, the domain of mass can be uniquely determined under certain constrains, and the domain inverse problem is a correct class of the inverse gravimetry. Under this correct class, we design the neural network for learning by mimicking the level-set formulation for the inverse gravimetry. Numerical examples illustrate that the method is able to recover mass models with non-constant density contrast.
Paper Structure (8 sections, 1 theorem, 9 equations, 4 figures, 1 table)

This paper contains 8 sections, 1 theorem, 9 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Learning on the correct class
  3. Network architecture and training
  4. Building training set on the correct class
  5. Network architecture
  6. Training and optimization
  7. Results
  8. Conclusion

Key Result

Theorem 2.1

Let $\Omega_0$ be a convex domain with analytic (regular) boundary, $\Sigma_0\subset\partial\Omega_0$ be a nonempty hyper-surface, and $\Omega\subset\Omega_0$ be a bounded domain with connected $\mathbf{R}^n\setminus\overline{\Omega}$. $D\subset\Omega$ denotes the domain of mass anomaly having piece

Figures (4)

  • Figure 1: Training set on the correct class. (a)-(d) show 4 examples of the 20,000 models we have built. The left column plots the mass distributions $\mu$, and the right column plots the simulated measurement data $\nabla U$ with $0-5\%$ Gaussian noises.
  • Figure 2: A modified U-net architecture for $\tilde{\Lambda}_\Theta(\cdot)$. The overall neural network for training is $\Lambda_\Theta(\cdot)=f\, \sigma(\tilde{\Lambda}_\Theta(\cdot) )$.
  • Figure 3: Results 1: The trained neural network $\Lambda_\Theta$ is implemented on the test set of 2000 samples. (a)-(d) show 4 examples of the 2000 test samples. The 1st column plots the measurement data with $0-5\%$ Gaussian noises, which are the inputs of $\Lambda_\Theta$; the 2nd column plots the recovered solutions; the 3rd column plots the true models.
  • Figure 4: Results 2: To test the generalization ability, the trained neural network $\Lambda_\Theta$ is implemented on the set of 200 salt models. (a)-(d) show 4 examples of the 200 salt models. The 1st column plots the measurement data with $0-5\%$ Gaussian noises, which are the inputs of $\Lambda_\Theta$; the 2nd column plots the recovered solutions; the 3rd column plots the true models.

Theorems & Definitions (1)

  • Theorem 2.1